Curves and share a common tangent, where .
What is the minimum value of ?
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As a → 1 , the curve y = a x approaches the line y = 1 ; and the curve y = lo g a x , i.e. the curve x = a y approaches the line x = 1 . These two lines obviously intersect at ( 1 , 1 ) , so for values of a very close to 1 the two curves will also intersect in the vicinity of ( 1 , 1 ) . Because the two curves are always concave up and concave down respectively, they will intersect at a second point in the first quadrant further away from the origin, and they will not be able to share a common tangent.
As the value of a increases, the curves will "bend away" from each other, and eventually they will be tangent to each other, i.e. intersect at a single point. The value of a at which this happens will be the minimum value at which the curves share a common tangent. For that value of a , since the curves are inverses of each other, the intersection point must lie on the line y = x , and in fact, the line y = x must be their common tangent.
Let the intersection point be ( α , α ) . At that point the function values will be equal, and the values of their derivatives will both be 1 . Then we have the system
a α a α ln a = lo g a α = α ln a 1 = 1 ( 1 ) ( 2 ) [ function values are equal ] [ derivatives are both equal to 1 ]
Substituting ( 1 ) into the first term in ( 2 ) , we get
ln a lo g a α ln α α = 1 = 1 = e [ using the rule ( lo g x y ) ( lo g y z ) = lo g x z ]
Then substituting this value of α into the second term in ( 2 ) , we get
e ln a 1 ln a a = 1 = e 1 = e e 1 ≈ 1 . 4 4 4 6 6 7 8 6